Solutions of Partial Differential Equations for Mean Molar...

6
This work has been digitalized and published in 2013 by Verlag Zeitschrift für Naturforschung in cooperation with the Max Planck Society for the Advancement of Science under a Creative Commons Attribution 4.0 International License. Dieses Werk wurde im Jahr 2013 vom Verlag Zeitschrift für Naturforschung in Zusammenarbeit mit der Max-Planck-Gesellschaft zur Förderung der Wissenschaften e.V. digitalisiert und unter folgender Lizenz veröffentlicht: Creative Commons Namensnennung 4.0 Lizenz. Solutions of Partial Differential Equations for Mean Molar Functions F. E. Wittig Institut für Physikalische Chemie der Universität München Z. Naturforsch. 34 a, 99—104 (1979); received November 4, 1978 Dedicated to Prof. Dr. G.-M. Schwab on his 80th birthday The well known formulas for computing the partial molar functions from a given mean molar function are treated as deferential equations for computing the mean molar function from any given partial molar function. Solutions do not depend on the number of components, but only on the choice of three indices: the index d of the dependent mole fraction xa to be eliminated prior to any computations, the index j of a pivot mole fraction xj and the index i of the partial molar function yi. An arbitrary number of additional mole fractions of the other components safe xa may be linked to the pivot mole fraction Xj. The simple solution: y = (xj — %) iy , yi = (xj öij) 2 Xfj and Xif = d Itj/dxj holds for an arbitrary number of components, if the (c — 2) mole fractions xi safe xa and xj are transformed to new variables found from the auxiliary equa- tions. Three different cases arise if either i = d, i = j or i =(= d, i =|= j is chosen. Formulas for the three sets are provided. As an example a simple interpolation formula for ternary systems is discussed. Previous experimental results on heats of mixing A M of liquid B-metal binary systems have been evaluated and discussed, using a so called ^-function [1,2] ÄM = aj2(l — z2(1) as suggested by Wagner [3]. This convenient method however failed in subsequent studies in ternary systems [4]. We had to apply rather intricate computer procedures to find formulas for Ä M and the three partial molar heats hi M . Therefore we tried to find simpler methods to process data in ternary systems with simple programmable desk calculators. A thorough study of pertinent formulas and methods seemed to indicate a missing link in the theory of such functions: Experimental values of excess chemical potentials FI,^ = RT In (y* = activity coefficient) in binary systems usually are evaluated by integration of the so called Gibbs- Duhem-equation [3]. In ternary systems integration is possible along particular paths of integration, e.g. X2IX3 = constant, as shown by Darken [3, 5, 6]. Another solution by Wagner [3] introduces besides X2 a new variable y = ns/(ni -f- W3) in the Gibbs- Duhem-equation. Our previous attempts to find reasonable formulas by trial and error always rendered functions with quotients of mole fractions. Therefore we supposed some hidden reason for the efficiency of such quotients. Reprint requests to Prof. Dr. Franz E. Wittig, Institut für Physikalische Chemie, Universität München, Theresien- straße 37, D-8000 München 2. Trouble with such functions seems to have a simple reason: Extensive functions Y— Y(T,p, ni) are homogeneous functions of the first degree of the c independent mole numbers ni (c number of com- ponents). Therefore we get with Euler's equation Y = ^myi with the definition yi = (3 F / 8 » , k ^ l (1) and the more useful differential dY = ^yldnl (2) with a short proof. (By partial differentiation to some njc of (1) we get: 2 n i (tyi/dnk) = 0 , 1 therefore too 2 2 n i fiyil&rik) dn* = 0 and k I ^nidyi = 0. 1 So the differential of (1) renders (2). The differentials dyi are more versatile, because they can be ex- panded with any set of appropriate variables of composition.) In practice, however, the mean molar func- tions y = y(T,p,xi) are studied, because the number of independent variables of composition is reduced to (c — 1) because of 2 xi = 1 and = (3) In this way labour with experiments and com- putations is reduced by an order of degree. But this results inevitabily in more trouble with formulas.

Transcript of Solutions of Partial Differential Equations for Mean Molar...

Page 1: Solutions of Partial Differential Equations for Mean Molar ...zfn.mpdl.mpg.de/data/Reihe_A/34/ZNA-1979-34a-0099.pdf · some njc of (1) we get: 2 ni (tyi/dnk) = 0 , 1 therefore too

This work has been digitalized and published in 2013 by Verlag Zeitschrift für Naturforschung in cooperation with the Max Planck Society for the Advancement of Science under a Creative Commons Attribution4.0 International License.

Dieses Werk wurde im Jahr 2013 vom Verlag Zeitschrift für Naturforschungin Zusammenarbeit mit der Max-Planck-Gesellschaft zur Förderung derWissenschaften e.V. digitalisiert und unter folgender Lizenz veröffentlicht:Creative Commons Namensnennung 4.0 Lizenz.

Solutions of Partial Differential Equations for Mean Molar Functions F. E. Wittig Institut für Physikalische Chemie der Universität München

Z. Naturforsch. 34 a, 99—104 (1979); received November 4, 1978

Dedicated to Prof. Dr. G.-M. Schwab on his 80th birthday The well known formulas for computing the partial molar functions from a given mean molar

function are treated as deferential equations for computing the mean molar function from any given partial molar function. Solutions do not depend on the number of components, but only on the choice of three indices: the index d of the dependent mole fraction xa to be eliminated prior to any computations, the index j of a pivot mole fraction xj and the index i of the partial molar function yi. An arbitrary number of additional mole fractions of the other components safe xa may be linked to the pivot mole fraction Xj. The simple solution: y = (xj — %) i y , yi = (xj — öij)2 Xfj and Xif = d Itj/dxj holds for an arbitrary number of components, if the (c — 2) mole fractions xi safe xa and xj are transformed to new variables found from the auxiliary equa-tions. Three different cases arise if either i = d, i = j or i =(= d, i =|= j is chosen. Formulas for the three sets are provided. As an example a simple interpolation formula for ternary systems is discussed.

Previous experimental results on heats of mixing AM of liquid B-metal binary systems have been evaluated and discussed, using a so called ^-function [1 ,2]

ÄM = aj2(l — z2)£ (1)

as suggested by Wagner [3]. This convenient method however failed in subsequent studies in ternary systems [4]. We had to apply rather intricate computer procedures to find formulas for ÄM and the three partial molar heats hiM. Therefore we tried to find simpler methods to process data in ternary systems with simple programmable desk calculators.

A thorough study of pertinent formulas and methods seemed to indicate a missing link in the theory of such functions: Experimental values of excess chemical potentials FI,^ = RT In (y* = activity coefficient) in binary systems usually are evaluated by integration of the so called Gibbs-Duhem-equation [3]. In ternary systems integration is possible along particular paths of integration, e.g. X2IX3 = constant, as shown by Darken [3, 5, 6]. Another solution by Wagner [3] introduces besides X2 a new variable y = ns/(ni -f- W3) in the Gibbs-Duhem-equation. Our previous attempts to find reasonable formulas by trial and error always rendered functions with quotients of mole fractions. Therefore we supposed some hidden reason for the efficiency of such quotients.

Reprint requests to Prof. Dr. Franz E. Wittig, Institut für Physikalische Chemie, Universität München, Theresien-straße 37, D-8000 München 2.

Trouble with such functions seems to have a simple reason: Extensive functions Y— Y(T,p, ni) are homogeneous functions of the first degree of the c independent mole numbers ni (c number of com-ponents). Therefore we get with Euler's equation

Y = ^myi with the definition yi = (3 F / 8 » , k ^ l (1)

and the more useful differential

dY = ^yldnl (2)

with a short proof. (By partial differentiation to some njc of (1) we get:

2 n i (tyi/dnk) = 0 , 1

therefore too

2 2 ni fiyil&rik) • dn* = 0 and k I ^nidyi = 0. 1

So the differential of (1) renders (2). The differentials dyi are more versatile, because they can be ex-panded with any set of appropriate variables of composition.)

In practice, however, the mean molar func-tions y = y(T,p,xi) are studied, because the number of independent variables of composition is reduced to (c — 1) because of

2 xi = 1 and = (3)

In this way labour with experiments and com-putations is reduced by an order of degree. But this results inevitabily in more trouble with formulas.

Page 2: Solutions of Partial Differential Equations for Mean Molar ...zfn.mpdl.mpg.de/data/Reihe_A/34/ZNA-1979-34a-0099.pdf · some njc of (1) we get: 2 ni (tyi/dnk) = 0 , 1 therefore too

F. E. Wittig • Solutions of Partial Differential Equations for Mean Molar Functions 100

First we have to find how to deal with the new functions and variables. Of course we can derive at once from (1) by division with the sum of number of moles n

y = ^ x i y i and dy = y y t d x i (4) (because from 2 n i tyi= too ^ x i tyi= 0)-

Such "symmetric" formulas, however, still com-prise a dependent mole fraction xa, that may be choosen arbitrarily from the c mole fractions in c different ways, and has to be eliminated prior to any computations by

xa = 1 — x i a n d dxa — — (5)

by separation of xd in (3) (the symbol ]> indicates, that the term with the index d has been omitted from the sum). From (4) we get

y = XdVd + xi Vi a n d

dy = yd xd + ^{d) Vi &xi (4 a)

and using (5) finally

y = yd + 2(<i) xi & ~ y*) and

dy = ^(d) _ yd) dxt . (6)

Clearly the structure of such "asymmetrical" formulas will be determined by the choice of the index d of the dependent mole fraction x^, to be eliminated prior to computations. From (6) we finally get the partial derivatives of the new func-tion y with respect to the new variables xi:

{dylfai)T)PtXm = yi-yd, m =M (7)

only rendering (c— 1) equations for the unknown c functions yi. Therefore we have to use (6) as additional equation, rendering

ya = y - ^ d ) x i { ^ y l ^ i ) (8) as shown by Haase [7, 8].

Clearly the relations between y and the yi are much more intricate than the simple formulas (1) for Y and yi in terms of the mole numbers rti.

Sometimes in physics the functions and variables found at first sight are not the most efficients ones with reference to the mathematics involved, as known from theoretical mechanics. Therefore we tried to find other functions and variables, render-ing at least one partial molar function as a simple derivative of a function of the molar function.

This problem will be solved by treating the equations for computing the partial molar functions by partial differentiation of the given mean molar

function as partial differential equations for the mean molar function, if some partial molar function yi is given. As new function we get the so called integral control function I i j in terms of a pivot mole fraction Xj and (c — 2) new variables qij, rij , of f i j , depending on the three different possibilities of choosing i = d, i = j, or i 4= d, i 4= j. The new variables, qij for quotient, r/y for ratio and f i j for fraction, are quotients of mole fractions, as pre-sumed before. The new functions and variables will be treated as shown with y in terms of the x with formulas (4 a), (6), (7) and (8), starting with the differential dI i j . Of course we shall meet the same trouble, because in this case too only (c— 1) partial derivatives can be found. This is only a question of the number of independent variables and not of the kind of variables. But we can find one equation rendering a function of a partial molar function as a simple partial derivative of I i j with respect to the pivot mole fraction Xj.

Later applications are possible without going through the subsequent expositions. Some simple applications and an interpolation formula for ternary system are provided on the last pages. The practical application has to be left to subsequent papers in view of the amount of computations in-volved. It will take some more papers to deal with the more intricate case of systems with electron transfer [4].

1. Control Functions

From (7) and (8) we get a formula for any partial molar function [4]

yi = y - to _ <*«) (9) using the Kronecker-symbol du = 0 for i =)= I and <5« = 1 for i = l. For any pure component i, given by Xi= 1, all other xi = 0, always yt = y holds. Therefore, any factor in the sum of (9) has to become zero for Xi = 1, all xi = 0. By this reason we get factors xi, but (xi — 1) in (9). In the binary case we get with xj = x\ or xj = x2 the general formula

Vi — y ix] öij) (dy/dxj). (10)

We first tried to get as simple functions and symbols as possible for binary systems. Obviously the structure of the formulas only depend from the indices i and j. Therefore the following functions

Page 3: Solutions of Partial Differential Equations for Mean Molar ...zfn.mpdl.mpg.de/data/Reihe_A/34/ZNA-1979-34a-0099.pdf · some njc of (1) we get: 2 ni (tyi/dnk) = 0 , 1 therefore too

101 F. E. Wittig • Solutions of Partial Differential Equations for Mean Molar Functions

and symbols were choosen [9, 10] y = — [Xj — dij) Ii) and

Vi = (X) — dij)2 Xij. (11)

Putting (11) to (10) the simple partial differential

Xi} = (8 Itj/dxj) (12)

results. X i j and I i j were called control function and integral control function (german: Formfunktion), because such functions carry the information on a particular function in some system, and control the shape of the graphs.

Darken [6] recast (10) to

Vi (Xj — dij)2 dxj \ (Xj — dij)

-y (13)

and called the left side function X i j "alpha" and "beta" function, but seemingly did not pay par-ticular attention to the function on the right side. The integral control function I i j is related to the apparent molar functions [9]. Clearly Darken's formula (13) is identical with our formulation (12), but obviously could not be applied to the multi-component case given by (9).

2. The Binary Case

Arranging (10) to

(Xj — dij) y' — y= — yi (14)

we get a simple differential equation to compute y from any given yi with the solutions

y = — (Xj — dij) C

dC/dxj = ytKxj - dij)2 . (15)

Clearly C = I i j , and the functions I i j and X i j are solutions of (14).

To get uniform symbols (1) has been replaced by

y™=-Xj(Xj-l)Xj (16)

but the control function Xj is not a solution of (14) and therefore only useful in binary systems.

3. The Multicomponent Case

Arranging (9) we get a partial differential equation

2 (xi - hi) (ty/fai) = y-yi (17)

for computing y from a given partial molar fun-tion yi. Lagrange's auxiliary equations, for con-

venience written in reverse order d y cLci &X2

y — yi Xi — da x2 — di 2 dxr.

Xc — die (18)

suggest the arbitrary choice of a pivot mole frac-tion Xj. (Of course, in (17) and (18) the dependent mole fraction xa is absent. If <Z= 1, d = 2 or d = c, such terms have to be omitted.)

Connecting first dy and dxj we arrive at

(xj — dij) dy = (y — yt) dx} (14a)

identical with (14). The solutions (11) and (12) of the binary case hold even with an arbitrary number of components, if the following solutions for the other (c — 2) independent mole fractions xi are taken in account. For all other mole fractions xi except xa and Xj we find simple proportionality from

(xi — da) = k(xj — dij). (19)

After fixing the indices d and j of Xd and Xj obviously three different sets of solutions arise by choice of the index i of the partial molar function yi.

3.1. The qij-set with i = d

Taking yi as yd, the Kronecker-symbols in (18) and (19) wall vanish rendering from (19)

xi = qij Xj . (20)

The limiting values of the so defined new variables qij are qij = 0 for xi = 0, but for Xj = 0 the qij become infinite in any subsystem not containing the component j. This may limit the practical use of this set.

3.2. The rij-set with i = j

Taking yi as yj the Kronecker-symbols dji will vanish, but of course djj = 1. From (19) follows

xi = r i j ( l — X j ) (21)

defining new variables rij with the convenient limit-ing values rij - 0 for xi = 0, and rij = 1 for xi = 1.

3.3. The fij-set with i =(= d, i =)= j

When choosing the index i different from d or j, e.g. i = f, the corresponding mole fraction Xf will appear in (18) or (19) in a bracket (xf— 1), whereas

Page 4: Solutions of Partial Differential Equations for Mean Molar ...zfn.mpdl.mpg.de/data/Reihe_A/34/ZNA-1979-34a-0099.pdf · some njc of (1) we get: 2 ni (tyi/dnk) = 0 , 1 therefore too

102

all other Kronecker-symbols will vanish. From (19) follows a new set of variables

F. E. Wittig • Solutions of Partial Differential Equations for Mean Molar Functions

4. The Differential d/, ,

(1 — Xf) = jf} Xj and xi = fij Xj . (22)

Subsequent computations can be reduced by first assessing the differential of I i j (11)

The fij will show the same limiting values as the qij, putting the same limitations on this set.

d L j = y dXj — (xj — dij) dy

(Xj — dij)2

Inserting (6) and arranging we get

d I i j = [ya + djj{yj — yd)] dxj + — yd) [xi dx} — (Xj — dy) ds;j {Xj — dij)2

(23)

(24)

In the binary case, any xi = 0 and dxi = 0, we get again the solution (11) and (12). With three and more components obviously (24) has to be reduced by judicious choice of new variables to arrive at fairly simple expressions. Any set of new variables can be tested by inserting in (24).

5. Formulas for the R-set

According to 3.2 we put i = j and djj= i. Inserting (21) and

dxi = (1 — Xj) drij — Uj dx} (25)

wet get from (24)

yj dXj + (Xj — l)2 (yi — yd) drij dljj =

(Xj - I)2

Defining the symbols

/ Mjj dxi

Xjj = Rij = 07 j j dri l j jxj,Ticj

(26)

, f c = H , (27) 3 /rij

we get the formulas

y} = (Xj — l ) 2 Xjj and yi~yd= Rtj. (28)

Unfortunately we get only one simple formula for y j , when using the convenient variables rij. For all other partial molar functions we have first to assess yd by inserting (21) and (28) in (6), arranging to

y = yd{ 1 — Xj) + Xj yj + 2 t W > » r l j R l j ( l - X j ) (29)

dividing by (1 — Xj), noting (3) and (28) and finally arriving at

yd = Ijj -Xj{ 1 - Xj) Xjj - 2 « * ' » rij Rij . (30)

Any other partial molar function besides y j and yd is found by

yk = yd + Rkj '31)

or fully

yk = Ijj — Xj (1 — Xj) Xjj

-Z(d'})(rij-dlk)Rj. (32)

6. Formulas for the Q- and i'-set

Proceeding in the same way we get the following definitions and formulas for use with the less convenient variables qij and fi j according (20) and (22)

Xdj = 07, dj CXi Qij

07, dj <\

Xj2 Xdj , yi

fylj )xj, qk}

yd — Qij yd

or fully

yi = Xj2 Xdj — Qij , yj= — Id] — Xj{ 1 — Xj) Xdj

+ 2«*-» qij Qij and for the T'-set

Mfj tej //„

(33)

(34)

(34 a)

(35)

Xfj = F f j J ^ L \ \ Zfn L in

Fu = dlfj_

yf = Xj2 Xfj , yd = Xj2 Xfj — Ff}, yi = Xj2 Xfj - Fn - Fij,

(36)

(37)

yi = - hi + xAxj - ! ) x f j + ( f f j - l ) F f j + Z(d'f'1)fijFv- (38)

The functions I i j follow from the definition (11):

y = — (xj — 1) Ijj, y = — Xj Idj,

y = — Xj Ifj (39) substituting the new variables from the R-, Q- or T'-set for the (c — 2) mole fractions xi besides the eliminated xd and the pivot mole fraction Xj.

Page 5: Solutions of Partial Differential Equations for Mean Molar ...zfn.mpdl.mpg.de/data/Reihe_A/34/ZNA-1979-34a-0099.pdf · some njc of (1) we get: 2 ni (tyi/dnk) = 0 , 1 therefore too

103 F. E. Wittig • Solutions of Partial Differential Equations for Mean Molar Functions

Formulas for the Q- and F-set are simpler, than for the .R-set. Therefore the Q-set may be useful, if the limiting behaviour of the variables qij for Xj = 0 is of no importance. The somewhat more complicated .F-set seems to offer no advantages at present.

7. Examples

Prior to computations in multicomponent sys-tems some attention should be paid to the judicious choice of the indices d, i, j and I. Experimental data of some ternary excess chemical potential /jl may be processed by choosing i = 1 ,2 or 3. But then in view of the advantages of the variables rij we should prefer j = i to apply the J?-set. After fixing i and j = i, we are free to eliminate one of both remaining mole fractions as the dependent variable xa. Then the remaining mole fraction is the xi, to be eliminated by rij. Putting e.g. i = 3, we take x3 as pivot mole fraction Xj. Then we may eliminate x\ = xa, and take as remaining xi, to be replaced by r23 = #2/(1 — £3). In this way we get the indices i = j = 3, d = 1 and I = 2. (In systems with c components we get (c —2) different mole fractions xi and therefore as much different indices I.) The numbers of indices have to be put into the general equations of the i?-set, e.g. X"33 = /M3E/(1 — x3)2 and dl33/dx3 = X33. Keeping X2IX3 constant in Darken's method in ternary systems clearly points to variables of the Q-set. In view of (20) we have xi = x2 and x j = x 3 . As X2 = <72323, this method means to replace X2 by <723, and to keep <723 constant. This means too, to eliminate x\ as the xa- Wagner's variable

y = nzKm + n3) - £3/(1 — x2)

is related to the R-set by choosing j = 2, I — 3 and therefore d= l . I n viewof(21) we get £3 = 7-32(1 — X2) and y = 7-32 in our system.

In ternary systems mean molar functions y can be represented by a power series expansion

= x * k x 3 1 • ( 4 ° ) k I Obviously x\ = xa or d= 1. Using (9) the following formulas for the three partial molar functions are found

!/I = 2J(1 — k — I) aki x2k xj , (41) k I

= — k — l)x2x3-\- k x3] k I

• akl X2k~1 X31'1 , (42)

= — k — l)x2x3 + lx 2] k I

• akt x2k~1 x3*-i. (43)

In this case, the coefficients of y± are simple multiples of the coefficients aki of y.

Choosing x3 as pivot mole fraction Xj, j = 3, and preferring the $-set for ease of computation we get

X2 = ?23 x3 . (44)

Putting (44) to (40) and dividing by (—x3) we get regarding (11) and (39)

/13 = - 2 2 ?23* (45) k I

and by differentiation with respect to x3 and <723 in view of (33)

Z i 3 = 2 2 - k ~ l) a*i x*k + l~ 2 ' <46) k I

Q23 — 2 2 ^ v™*'1 X 3 k + l ~ 1 • ( 4 ? ) k I

Putting 113, X13 and Q23 in (34) and (35) and finally again substituting X2 for qz3x3 the formulas (41), (42) and (43) are obtained.

8. A Simple Interpolation Formula for Ternary Systems

The rather lengthy and tedious computations for assessing the matrix Aki in (40) from experimental data may be considerably reduced by first comput-ing approximate data from the formulas for the binary subsystems using a simple interpolation formula. According to our experience with ternary metallic systems even simple interpolation formulas render about 90% of the experimental values. As the precision of such data only seldom approaches 1%, the precision of the residue will at best approach 10%. Therefore a simple additional procedure for assessing a matrix for the residue will suffice as a rule.

To assemble interpolation formulas for systems with c components from the formulas for the binary subsystems we have to use some additional indices to indicate different systems. Tentatively, — we have still to gather more experience —, the follow-ing method is adopted: the numbers of the com-ponents are indicated in an additional index. The first number is the index of the mole fraction not used in the formula. In subsystems without the

Page 6: Solutions of Partial Differential Equations for Mean Molar ...zfn.mpdl.mpg.de/data/Reihe_A/34/ZNA-1979-34a-0099.pdf · some njc of (1) we get: 2 ni (tyi/dnk) = 0 , 1 therefore too

104 F. E. Wittig • Solutions of Partial Differential Equations for Mean Molar Functions

component d this will be another mole fraction than xa-

In a ternary system y123 means the formula for the mean molar function y in terms of and x$. y2Z is the formula for the binary system with the components 2 and 3 in terms of X3.

Choosing x\ as the dependent mole fraction and X3 as the pivot mole fraction Xj, and finally putting X2 — T23 (1 — xs) according to (21) the follow-ing simple formula

y123 = y13 + r2s(y23 - y13) + (1 - * 3 ) 2y 1 2 (48)

allows for linear interpolation between the binary systems (13) and (23), whereas the third system (12) is interpolated by multiplying with (1 — X3)2, as done previously by Kohler [10].

In y12 the variable x2 has to be substituted by r23

to stay within the limiting values 0 and 1 of x2 . To get the formulas for the three partial molar

functions the following procedure is applied: 1. According to (39) we get by division with

(1 - **)

= HI + r 2 3 (I 2 l - Jg) + (1 - *.) y12 . (49) 2. Differentiation with respect to and r23

renders

Z g 3 = Z g + 'as ( X U - Z g ) - y™ , (50)

= f g - i g + ( ! - * » ) fo12)' (51)

with (y12)' = d?/12/dr23.

[1] F. E. Wittig, Z. Elektrochem. 63, 327 (1959). [2] F. E. Wittig and G. Keil, Z. Metallkunde, 54, 576

(1963). [3] C. Wagner, Thermodynamics of Alloys, Addison-

Wesley Press, Inc., Cambridge 42, Mass. 1952. [4] F. E. Wittig, N. Saes and W. Waldherr, Rev. Chim.

Miner. 9, 71 (1972). [5] L. S. Darken, J. Amer. Chem. Soc. 72, 2909 (1950).

3. yi as yd follows from (30) using (49), (50) and (51) as

y i = i g - xt(i - x3)[X\l+ r 2 3 ( Z I - z g ) ] + (1 - X32)y™ - r23( 1 - X3)(y12)'. (52)

4. According to (28) y2 is found by simply adding (51) to V l

y2 = I23 ~ *3(1 - x3)[Xll + r 2 3 (X| - Z g ) ] + (1 - X32) y12 - (r23 - 1) (1 - x3) (yi2)'.

(53)

5. y3 is found by multiplying (50) with (1 —X3)2

according to (28). Two different checks for such formulas may be

applied: 1. Putting the variables equal 0 or 1 and checking

the borderline behaviour. Putting e.g. x3 = 0 (52) reduces to (7g = 0 for ar3 = 0)

yi = y12 — r23 (di/12/dr23) (52 a)

being the formula for y\12 in terms of r23. 2. Putting the three partial molar functions in (6)

formula (48) will be found. Applications to evaluations of heats of mixing

and of molar free energies of mixing in ternary systems will be presented in subsequent papers. As long as the functions for the binary systems may be represented by power series, the formulas (48) — (53) provide fairly simple rules for assembling matrices for the ternary functions.

[6] L. S. Darken, R. W. Gurry, and M. B. Bever, Physical Chemistry of Metals, McGraw-Hill Book Company, Inc., New York 1953.

[7] R. Haase, Z. Naturforsch. 3a, 285 (1948). [8] R. Haase, Thermodynamik der Mischphasen, Sprin-

ger-Verlag, Berlin 1956. [9] F. E. Wittig, Chemie-Ing. Technik 42, 1037 (1970),

43, 1211 (1971). [10] F. Kohler, Monatsh. Chem. 91, 738 (1960).